A241555 Triangle read by rows: Number T(n,k) of 2-colored binary rooted trees with n nodes and exactly k <= n nodes of a specific color.

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 11, 16, 11, 3, 6, 26, 50, 50, 26, 6, 11, 60, 143, 188, 143, 60, 11, 23, 142, 404, 656, 656, 404, 142, 23, 46, 334, 1105, 2143, 2652, 2143, 1105, 334, 46, 98, 794, 2995, 6737, 9934, 9934, 6737, 2995, 794, 98, 207, 1888, 7999, 20504, 35080, 41788, 35080, 20504, 7999, 1888, 207

Offset: 0

Author

David Serena, May 17 2014

Keywords

• nonn
• tabl

Comments

T(n,k) = T(n,n-k) by definition.

First column is A001190.

Row sums are given by A226909.

Examples

			Triangle begins:
   1;
   1,   1;
   1,   2,   1;
   2,   5,   5,   2;
   3,  11,  16,  11,  3;
   6,  26,  50,  50,  26,   6;
  11,  60, 143, 188, 143,  60,  11;
  23, 142, 404, 656, 656, 404, 142, 23;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274
• David Serena and William J. Buchanan, Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions, arXiv:2503.02663 [math.CO], 2025. See p. 17.

Crossrefs

Cf. A001190, A226909.

Programs

Mathematica

B[m_] := Module[{u}, u = Table[0, {m}]; u[[1]] = 1; For[n = 1, n <= Length[u] - 1, n++, u[[n + 1]] = (1 + y)*(Sum[u[[i]]*u[[n + 1 - i]], {i, 1, n}] + If[OddQ[n], u[[Quotient[n, 2] + 1]] /. y -> y^2, 0])/2]; u];
CoefficientList[#, y]& /@ B[11] // Flatten (* Jean-François Alcover, Sep 24 2019, from PARI *)

PARI

B(n)={my(u=vector(n)); u[1]=1; for(n=1, #u-1, u[n+1]=(1+y)*(sum(i=1, n, u[i]*u[n+1-i]) + if(n%2, subst(u[n\2+1], y, y^2)))/2); u}
{ my(A=B(10)); for(n=1, #A, print(Vec(A[n]))) } \\ Andrew Howroyd, May 21 2018

Extensions

Edited by Nathaniel Johnston, Sep 11 2014

Missing term inserted and a(45) and beyond from Andrew Howroyd, May 21 2018